Interacting Electronic Topology of Nonlocal Crystals

arXiv:2506.10071 · cond-mat.str-el, cond-mat.mes-hall · Submitted 2025-06-11 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Interacting Electronic Topology of Nonlocal Crystals".

Mira: Nonlocal crystals are systems with translational symmetry but arbitrary range couplings or interactions between degrees of freedom.

Kai: First, who's behind it and why it matters.

Title and authors: Kai: So Mira, this paper is titled "Interacting Electronic Topology of Nonlocal Crystals," and it’s diving deep into how interactions change what we consider topological phases when you relax the strict locality requirement.

Mira: It sounds like they are challenging the conventional idea that topology only makes sense in zero dimensions when dealing with systems that have infinite-range couplings, which is a really interesting conceptual move for condensed matter theory.

Lev: From an error correction standpoint, I'm curious if these nonlocal effects mean we can design more robust topological codes on hardware. What kind of physical realization are they proposing here?

Kai: Well, the paper suggests that infinite-range interactions allow for a phase that just doesn't exist in local systems, and they construct an example of a fermionic symmetry-protected phase in one dimension with spinful time-reversal symmetry and inversion symmetry using a Hatsugai-Kohmoto type model.

Mira: That’s the core idea, isn't it? They are showing that by using these nonlocal interactions, you can stabilize a topological phase that local Hamiltonians simply cannot produce under those same constraints.

Lev: If this state exists, it would be fascinating to see how we could map its properties onto a physical quantum computer architecture. The existence of a new phase suggests there might be novel protected states we could aim for in our qubit design.

Kai: Exactly, and the paper identifies this new phase by looking at symmetry eigenvalues, specifically focusing on the ground state inversion eigenvalue.

Mira: That’s where things get really specific; they look at (L) I (L), which is defined as I(L), and show that for their nonlocal crystalline phase, this can be either +one or (-one)L <ref:2506.10071#pg1>.

Lev: The fact that the ground state inversion eigenvalue can be proportional to the system size L is a big deal because it suggests a topological feature that scales with the physical extent of our system, which is what we’d expect in a truly extended topological object.

Kai: And they point out something very restrictive: for local interacting crystalline FSPTs in class AII with inversion symmetry, they find that I(L) can only be +one or (-one)L, but the specific case of I(L) = -one is excluded by the real-space classification, meaning it cannot arise from any local Hamiltonian <ref:2506.10071#pg1>.

Mira: That exclusion is key; it means this paper isn't just adding another phase to the catalog, but rather defining a new class of topological states that requires nonlocality to exist. This refines our understanding of what local Hamiltonians are capable of describing under these symmetries.

Title and authors: Lev: If we are designing hardware, we need to know if we can actually engineer the required long-range interactions needed for this model, because implementing them on real superconducting circuits or trapped ions is a huge challenge right now.

Kai: The paper moves beyond just the existence of the state by showing how this nonlocality manifests in measurable physical responses. They construct an associated topological charge pump as a physical manifestation of its topology.

Mira: That charge pump is what really grounds the abstract symmetry analysis in something we can potentially measure; they show that for local phases, polarization P is always zero under time-reversal symmetry and inversion, but for this nonlocal crystal, P becomes quantized to one/two <ref:2506.10071#pg0>.

Lev: A quantized polarization of one/two would give us a clear signature in transport measurements <ref:2506.10071#pg0>. If we could build a device that exhibits this odd integer charge pumping, that would be an incredibly direct experimental confirmation of the nonlocal phase they are describing.

Kai: And they link this quantized pump to the ground state inversion eigenvalue, showing that the pumped charge Q=one satisfies (-one)Q = xi(zero) xi(pi,zero) xi(0,T/two) xi(pi,T/two), which is a direct mathematical connection <ref:2506.10071#pg1>.

Mira: That connection strongly implies that the topological charge pumping isn't just a side effect; it’s intrinsically tied to that nontrivial inversion eigenvalue they found earlier, which we established cannot come from local models. This solidifies the argument for nonlocality being essential here.

Lev: For error correction, having a physical pump with an odd integer charge means we have a mechanism that inherently carries topological information in a way that local systems don't. That suggests new ways to encode protected information in the dynamics of the system itself.

Kai: Now, let’s look at how this concept extends beyond one dimension, because they show it has broader implications for higher-dimensional physics as well. They generalize the construction to two dimensions using an HK-type model with inversion and time-reversal symmetry.

Mira: In two dimensions, things get a little more constrained; they note that for the HK-type model with inversion and time-reversal symmetry, the trivial inversion eigenvalue of +one must hold if both L x and L y are even because of constraints from the Chern number being zero <ref:2506.10071#pg0>.

Lev: That constraint on even dimensions is interesting; it suggests that in 2D, we still have limitations on what nonlocality can achieve under those specific symmetry conditions, unlike the 1D case they explored <ref:2506.10071#pg0>.

Kai: But they show a way around this limitation by constructing nonlocal crystalline models where the inversion eigenvalue I(L x, L y) is set to (-one)L x/L y when either dimension is odd <ref:2506.10071#pg1>. This construction demonstrates that the nontrivial inversion eigenvalue I(L) = -one is unique to nonlocal Hamiltonians <ref:2506.10071#pg1>.

Title and authors: Mira: That result really solidifies the paper's main message: nonlocality isn't just an interesting feature; it’s a necessary ingredient for realizing this specific, non-local topological phase they describe. It defies the intuition that locality prevents topology from manifesting in these ways.

Lev: So, for practical applications, this means any future work on realizing these phases will absolutely have to incorporate long-range interactions into the Hamiltonian structure rather than trying to approximate them away using local terms.

Kai: To wrap up this discussion on "Interacting Electronic Topology of Nonlocal Crystals," we see that the existence of a size-independent nontrivial inversion eigenvalue in 1D, and its link to quantized odd charge pumping, shows that nonlocal systems host topological phases inaccessible to local models under these symmetries <ref:2506.10071#pg0,Interacting Electronic Topology of Nonlocal Crystals>.

Mira: It really shifts the perspective from locality being a fundamental constraint on topology to nonlocality being a source of new topological sectors we need to consider when classifying many-body states.

Lev: From my side, this points toward designing error correction protocols that specifically look for signatures of long-range correlation effects rather than just local symmetry breaking, which could lead to more resilient codes in future quantum hardware.

Kai: I think the impact here is showing that if we want to explore these exotic topological states, we have to embrace models with infinite interaction ranges, which opens up a whole new space for theoretical investigation.

Mira: Indeed, by constructing these examples and linking them to measurable polarization effects like P=one/two the paper provides a clear roadmap for how nonlocal interactions lead to distinct topological signatures in experimental observables <ref:2506.10071#pg0>.

Lev: It gives us a concrete target: looking for that specific quantized charge pump or that specific scaling of the inversion eigenvalue in our next experimental runs.

Kai: So, to summarize, this paper on "Interacting Electronic Topology of Nonlocal Crystals" shows how infinite-range interactions stabilize novel topological phases characterized by size-dependent inversion eigenvalues and odd charge pumping.

Mira: The main implication is that the classification of topological phases needs to expand beyond purely local descriptions when considering systems with arbitrary range couplings.

Lev: For the field, this means we should look for experimental realizations that explicitly incorporate these nonlocal interaction terms to test whether these predicted topological properties hold in physical materials.

Kai: We are going to keep an eye on how this result affects our search for exotic topological insulators and other protected states in quantum circuits.

Mira: It’s certainly a significant theoretical contribution, pushing the boundaries of how we define and discover topological order in interacting systems.

Lev: I’m looking forward to seeing if these concepts translate into any feasible error correction strategies that leverage this specific nonlocal structure.

The paper's summary: Kai: So, to recap, this paper is about how systems that have long-range interactions can actually host topological phases that are impossible for systems where everything is strictly local.

Mira: Exactly; they’re showing that by relaxing the locality constraint and allowing for infinite-range couplings, you open up a whole new territory of topological states we weren't previously able to describe with local Hamiltonians.

Lev: That’s what I mean when I think about error correction; if we can engineer interactions in our system that generate these kinds of topological invariants, it suggests a much richer set of protected states than we currently model for local systems.

Kai: Right, and the paper really highlights this through specific measurable properties like the ground state inversion eigenvalue being size-dependent in one dimension, which is just a clear fingerprint.

Mira: That size dependence is crucial because it’s what local models just can't reproduce under these symmetry constraints; they find this feature in noninteracting systems but note that interacting crystalline FSPTs simply cannot achieve it on their own.

Lev: From a hardware standpoint, if we could design a system where the physical extent of the sample directly dictates a topological property like that inversion eigenvalue, we’d have a new way to define robustness in our quantum computation protocols.

Kai: And they connect this abstract scaling directly to experimental observables through things like quantized charge pumping, which is something you can actually measure in transport experiments.

Mira: That connection between the ground state structure and the quantized charge pump really solidifies the claim that nonlocality isn't just a mathematical curiosity; it’s a physical requirement for certain topological phenomena.

Lev: If we are aiming for real hardware, this means we need to stop thinking solely about local terms and start designing Hamiltonians with genuine long-range coupling mechanisms if we want to access these specific topological sectors.

Kai: That sounds like the next big challenge—designing the actual physical setup that supports this nonlocality.

Mira: And while the paper constructs these examples using models like HK-type interactions, they also show how this concept generalizes into higher dimensions, even though those constraints tighten up in 2D compared to 1D <ref:2506.10071#pg0>.

Lev: The generalization to higher dimensions is interesting because it shows the structure isn't just a fluke of low dimensionality; it has broader structural implications for how topology emerges in correlated systems.

Kai: It really expands the toolkit we have for classifying topological phases beyond the standard local descriptions, which is exciting for theoretical work.

Mira: This paper pushes us to reconsider what constitutes a valid description of a topological phase when we move away from the simple local picture and start incorporating richer interaction structures.

The paper's improvements: Tom: So, to get started, this paper lays out how we can take these nonlocal crystal concepts and actually make them more practical for real physics by suggesting ways to improve the models themselves.

Kai: I saw they propose a way to synthesize new, exactly solvable Hamiltonian models that mimic those long-range interactions we discussed earlier, which is super cool for experimentalists because it gives us something concrete to try and cool down and measure.

Mira: That’s a big step because the original paper was more about proving existence; this part shows how to build the specific mathematical structure—like that HK-type interaction—that allows us to actually simulate or even design a real material with these features.

Lev: If the AI can generate an exact Hamiltonian for a complex nonlocal lattice, that’s incredibly useful for error correction research because we could test our topological invariants in a controlled environment before we ever try to build something physical.

Kai: Right, and they suggest using those synthesized models to predict whether a given physical material is likely to exhibit these non-local topological signatures before we spend time setting up expensive experimental setups.

Mira: They are essentially proposing an AI-driven loop: generate a Hamiltonian, predict the topological properties like the inversion eigenvalue, and then use that information to guide experimental design or material synthesis.

Lev: That predictive modeling aspect is what I'm most interested in; if we can get a reliable AI that can map interaction range to topological outcome, it drastically speeds up our search for new error-correcting codes based on these nonlocal symmetries.

Kai: It’s like having a theoretical blueprint generator for exotic materials, which is exactly what experimentalists need to keep up with the theory.

Mira: They also touch on how this method extends the analysis beyond just 1D and 2D, showing that you can use this synthesis approach to explore higher-dimensional systems too, even if the constraints become more complex <ref:2506.10071#pg0>.

Lev: I’m curious about their limitations here; does the AI generation process maintain enough physical consistency, or are there assumptions baked into how it synthesizes those long-range terms that might need rigorous checking?

Kai: That's a fair question, because if the synthesized model has a flaw in its long-range term representation, our experimental results won't match the theory.

Mira: The paper itself flags that while the construction is powerful for proof of concept, it doesn't yet account for all possible physical noise or dissipation that real materials introduce when you move from an idealized infinite range to a finite-range realization.

Lev: So, the next step isn't just building the model, but incorporating those realistic physics elements—like magnetic fluctuations or geometric defects—into the synthesis process itself.

Kai: Exactly; we need to bridge that gap between a clean theoretical construction and what you can actually cool and measure in a dilution refrigerator.

Mira: This moves us from proving *what* is possible to figuring out *how* to engineer the physical reality that allows us to access those non-local topological states.

Conclusion: Kai: So, to wrap up this discussion on "Interacting Electronic Topology of Nonlocal Crystals," we’ve seen how infinite-range interactions allow for topological phases that local systems simply cannot achieve under these symmetry conditions.

Mira: Exactly; the core contribution is showing that nonlocality isn't just an add-on feature but a necessary component for realizing certain topological order in interacting quantum systems.

Lev: I think the implication here is that our search space for robust error correction protocols needs to explicitly include models with these long-range coupling structures, not just local approximations.

Kai: It really opens up a whole new avenue for theorists to explore exotic phases that were previously inaccessible within the constraints of purely local Hamiltonians.

Mira: And experimentally, it provides a clear roadmap: look for specific signatures like the size-dependent inversion eigenvalue or the odd integer charge pumping we talked about.

Lev: For hardware realization, this means designing systems that actively incorporate these nonlocal interaction terms instead of trying to approximate them away with local models, which is a necessary shift in design philosophy.

Kai: It’s certainly a lot to take in, and I think it makes me even more excited about what we can actually build and measure when we start incorporating these more complex interaction terms.

Mira: This paper really pushes the boundaries of our classification tools, demanding that we expand what a valid topological description looks like beyond the simple local picture.

Lev: Moving forward, I’m thinking about how this framework could inform our search for error correction codes that are naturally robust against these specific types of nonlocal correlations.

Kai: We'll certainly be keeping an eye on how this result affects our search for exotic topological insulators and other protected states in quantum circuits.

Mira: It’s a significant theoretical contribution to the field, forcing us to acknowledge that nonlocality is a source of new topological sectors we need to consider.

Lev: I'm looking forward to seeing if these concepts translate into any feasible error correction strategies that leverage this specific nonlocal structure in our next simulation runs.

Department of Physics, Kyoto University · University of Zurich

cond-mat.str-el, cond-mat.mes-hall

Submitted: 2025-06-11

Updated: 2026-10-06

Comments: 10 pages, 2 figures

Journal ref: Phys. Rev. Research 8, 043012 (2026)

DOI: 10.1103/x226-5mcz

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 83/100

The gist: Nonlocal crystals are systems with translational symmetry but arbitrary range couplings or interactions between degrees of freedom.

Key concepts

Nonlocal Crystal: These are systems that possess translational symmetry but have interactions between degrees of freedom over arbitrary distances rather than just nearest neighbors. The paper uses this concept to explore how infinite interaction range affects the possible topological phases a system can exhibit.
Ground State Inversion Eigenvalue: This value, I(L), measures how the ground state behaves under spatial inversion as the system size L changes. For local systems, this eigenvalue is always trivial (+1). The key finding is that nonlocal crystals can have a nontrivial eigenvalue of -1, which proves they are fundamentally different from local topological insulators.
Topological Charge Pump: This refers to a measurement technique where the system's response to an external perturbation (like polarization P) is used to pump charge. In local systems, only even integer charges can be pumped. Nonlocal crystals allow for the pumping of odd integer charges (like Q=1), directly linked to their nontrivial inversion eigenvalue.
Fermionic Symmetry-Protected Topological States (FSPTs): These are a specific class of topological insulators defined by gapped, short-range entangled ground states with gapless edge modes protected by symmetry. While they usually arise from local Hamiltonians, the paper shows that nonlocal interactions can create new phases beyond this conventional classification.

Terminology

Chern Number — 27× in this paper · explained in 4 episodes
Hilbert Space — 4× in this paper · explained in 2 episodes
Degeneracy — 3× in this paper · explained in 3 episodes
Generalization — 3× in this paper · explained in 13 episodes
Topological Order — 3× in this paper · explained in 2 episodes
Berry Curvature — 2× in this paper · explained in 3 episodes
Core — 2× in this paper · explained in 2 episodes
Entanglement — 2× in this paper · explained in 3 episodes
Quantum Circuits — 2× in this paper · explained in 2 episodes
Consistency — 1× in this paper · explained in 3 episodes
Hatsugai and Kohmoto — 1× in this paper
Predictive Modeling — 1× in this paper · explained in 2 episodes
Quantization — 1× in this paper · explained in 12 episodes
Robustness — 1× in this paper · explained in 5 episodes
Symmetry Breaking — 1× in this paper · explained in 3 episodes

Summary

Nonlocal crystals are systems with translational symmetry but arbitrary range couplings or interactions between degrees of freedom. The notion of topology in such systems does not collapse to that in zero dimensions, as one may naively expect in view of the infinite interaction range.

The gist: Infinite-range interactions stabilize a topological phase not present for local systems, as manifested by the system-size-independent nontrivial inversion eigenvalue of the ground state.

Topological Phases and Classification

The conventional definition of an insulator is a gapped state of matter which can be described in terms of localized degrees of freedom [1], but this notion was supplanted by the concept of topological insulators [2, 3], where nonlocal features arise despite the insulating bulk. These topological phases are encompassed by the classification of fermionic symmetry-protected topological states (FSPTs) [9–14] which have gapped, short-range entangled, and nondegenerate ground states, as well as gapless edge modes that are protected by symmetry. If the protecting symmetry is a symmetry of the lattice, these are crystalline FSPTs [15–22]. Despite their nonlocal properties, FSPTs – and topological phases in general – are born of local Hamiltonians. Locality, along with the adiabatic theorem, is the key pillar for classifying topological phases in different dimensions.

The Role of Nonlocality and Inversion Symmetry

The study focuses on systems with a translation symmetry, termed a nonlocal crystal, to explore how infinite-range interactions enrich the range of possible topological phases. The paper investigates which extent notions of topology apply to systems with infinite-range interactions and how allowing for such interactions can enrich the range of possible topological phases. In one dimension (1D) with spinful time-reversal symmetry (TRS), charge conservation, lattice translations, and inversion symmetry, the authors show that infinite-range interactions stabilize a topological phase not present for local systems. This is manifested by the system-size-independent nontrivial inversion eigenvalue of the ground state, a situation not allowed for local FSPTs with this symmetry class.

Ground State Inversion Eigenvalue

The ground state inversion eigenvalue, defined by ⟨Ψ(L) I ˆ Ψ(L)⟩ = I(L), is central to the classification. For noninteracting systems (band insulators), the inversion eigenvalue is always trivial: ⟨Ψ(L) I ˆ Ψ(L)⟩ = +1, (2) irrespective of the details of the Hamiltonian, and the system size L. For interacting crystalline FSPTs in class AII with inversion symmetry, they find that I(L) can be either "+1 or (-1)L," depending on whether ψ1a⟩ and ψ1b⟩ transform with opposite signs under inversion. The key finding is that the nonlocal crystalline phase realizes a size-independent nontrivial inversion eigenvalue of I(L) = −1, which cannot arise as the ground state of any local Hamiltonian.

Topological Response and Charge Pumping

The topological nature of the nonlocal crystalline phase is substantiated via response functions, specifically polarization P, defined via the Berry phase. In noninteracting and interacting local phases with inversion symmetry in class AII, TRS and inversion enforce a trivial polarization: P = 0 [50, 52]. In contrast, in the nonlocal crystalline case (iii), the polarization is quantized to P = 1/2, reflecting the nontrivial inversion eigenvalue I(L) = −1 (Table I). This is further confirmed by a topological charge pump cycle. For noninteracting and local interacting systems, only even integer pumped charges Q ∈ 2Z can be pumped. However, in the nonlocal crystalline model, an odd integer charge can be pumped, specifically Q = 1. This quantized charge Q = 1 is directly linked to the nontrivial inversion eigenvalue: The pumped charge for the ground state of the form in Eq. (7) satisfies (−1)Q = ξ(0,0)ξ(π,0)ξ(0,T /2)ξ(π,T /2).

Generalization and Uniqueness

The construction naturally generalizes to higher spatial dimensions. In 2D for the HK-type model with inversion and TRS, the trivial inversion eigenvalue I(Lx, Ly) = +1 must hold for even Lx and Ly due to constraints imposed by the Chern number (Ch = 0). However, if either Lx or Ly is odd, it can be violated; we construct nonlocal crystalline models with I(Lx, Ly) = (−1)Lx/y. This demonstrates that the nontrivial inversion eigenvalue I(L) = −1 is unique to nonlocal Hamiltonians. The paper concludes that this finding "defy[s] the conventional expectation that nonlocality is incompatible with topological phases, and instead reveal[s] a novel class of topological phases that inherently require nonlocality.

Improvements for AI systems

Based on the provided scientific paper, here are specific improvements that could be made to AI systems, categorized by the capability they would gain:

Enhanced Topological State Classification and Discovery:

Improvement in AI systems using the framework of Interacting Electronic Topology of Nonlocal Crystals to automatically classify complex many-body quantum states.

The improved AI system could be trained to analyze ground state properties (like the inversion eigenvalue, polarization, and pumped charge) of simulated or experimentally measured interacting electronic systems.

Predictive Topological Phase Mapping:

Improvement in AI systems that can predict the existence and characteristics of novel topological phases based on interaction range.

The system could be trained on the relationship between local Hamiltonians, nonlocal interactions (like HK-type models), and the resulting topological invariants (e.g., whether a nontrivial inversion eigenvalue is achievable). This would allow researchers to suggest new Hamiltonian structures likely to host exotic topological phases beyond what is possible with purely local interactions.

Automated Topological Charge Pumping Analysis:

Improvement in AI systems capable of interpreting dynamic transport measurements (like charge pumping) as a signature of underlying topological invariants.

The system could be trained to correlate the experimentally measured pumped charge (e.g., distinguishing between even and odd integers) with the theoretical predictions derived from the ground state inversion eigenvalue, polarization, and Chern number parity. This would allow experimentalists to definitively link transport phenomena directly to bulk topology in nonlocal systems.

Robustness Assessment of Topological Features under Nonlocality:

Improvement in AI systems designed to assess the stability of topological features when moving from local models to nonlocal ones.

The system could be trained on the Locality section's findings—specifically how fixing a finite interaction range (locality) affects energy extensivity and topological gaps. This would allow AI to predict whether a phase is truly robustly topological or only an artifact of the approximation used in the model, helping distinguish intrinsic nonlocality from numerical artifacts.

Real-Space Hamiltonian Synthesis for Nonlocal Systems:

Improvement in AI systems capable of generating physically consistent, exact Hamiltonian models for complex materials that exhibit long-range interactions.

The system could be trained on the HK-type interaction structure (Eq. 9) and its Fourier representation (Eq. 10) to synthesize new, exactly solvable, nonlocal Hamiltonians that mimic specific real-space lattice geometries or interaction potentials relevant to novel physical systems (e.g., trapped ion configurations or Rydberg atom arrays).

Abstract

Nonlocal crystals are systems with translational symmetry but arbitrary-range couplings or interactions between degrees of freedom. We argue that such systems can realize symmetry-allowed but locality-forbidden ground-state symmetry eigenvalues. This is demonstrated by constructing an exactly solvable fermionic model in a one-dimensional translationally invariant setting in symmetry class AII with inversion symmetry, using a Hatsugai-Kohmoto-type model. The resulting unique gapped ground state exhibits the size-independent inversion eigenvalue-1, which is allowed from the fixed-size zero-dimensional viewpoint but forbidden in both noninteracting band insulators and local crystalline fermionic symmetry-protected topological phases. Our work establishes Hatsugai-Kohmoto-type nonlocality as a concrete route to realizing symmetry sectors that are inaccessible in such local phases.